Why is saxophone growling produced by modulation of the sound waves?

[Daniel Petka]

TL;DR Summary

Playing any note on a saxophone and singing some other note produces a subharmonic = difference frequency, which hints that the 2 signals are modulated (product rule) Why does modulation occur? This is not trivial as it only works on reed instruments

My clarinet teacher once showed me a trick: you can play any note and then sing a fifth above that note and it will create the illusion of sounding an octave deeper. On a different sub, I asked about this technique:



It turns out that this is called saxophone growling. And it's no coincidence that my teacher played the sax. But here is the twist: it's not an illusion. I measured a spectrogram of three signal and there was an actual frequency one octave below the clarinet frequency. So it's not the missing fundamental effect, the fundamental frequency is there. Another observation: playing the third will actually produce two sinusoids: one at 2/3 the clarinet frequency and one at 1/3. So one thing clicked: this is amplitude modulation, but I'm struggling to understand why the signal is modulated. I get that it has to be nonlinear in some way but that is not too satisfying for me. Googling a bit about undertones I came across the duffing equation. A nonlinear system, in out case the voice + the vibrating clarinet reed can be driven below its resonance frequency. But I don't understand how this ties up to the modulation I have observed. Thanks for any ideas! I'm just an EE, so I struggle a bit when things aren't linear

 

[berkeman]

TL;DR Summary: Playing any note on a saxophone and singing some other note produces a subharmonic = difference frequency, which hints that the 2 signals are modulated (product rule) Why does modulation occur?

Interesting. As you probably know, mixing can only happen when there is some non-linear process involved. So my guess is that your singing the note drives the reed farther than the normal mechanical excursion, which provides the opportunity for this mixing to happen.

You can test this by varying the levels of your blowing and your singing. If you don't hear the sub-harmonic difference frequency mixing when you blow lightly and sing quietly, and you hear it much more when you play a fortissimo note and sing loudly, then you are driving the reed into its non-linear harmonic motion region.

Let us know what you find!

 

[Baluncore]

I don't think you have to look far to find a non-linearity in a saxophone. The reed is functioning as a switching diode, that rectifies the waveform, and so adds even harmonics to the sound of the instrument. The reed is operating between the mouth and a resonator, which colours the harmonic content.

It seems to me, that growling is actually a linear beat between the vocal, and the saxophone fundamental. That is why the growl frequency can be below both the voice and the fundamental. If the growl then interacts with the reed, it adds more harmonics.

 

[Daniel Petka]

Interesting. As you probably know, mixing can only happen when there is some non-linear process involved. So my guess is that your singing the note drives the reed farther than the normal mechanical excursion, which provides the opportunity for this mixing to happen.

You can test this by varying the levels of your blowing and your singing. If you don't hear the sub-harmonic difference frequency mixing when you blow lightly and sing quietly, and you hear it much more when you play a fortissimo note and sing loudly, then you are driving the reed into its non-linear harmonic motion region.

Let us know what you find!

Nope, I can make the clarinet growl even while playing piano

 

[Daniel Petka]

I don't think you have to look far to find a non-linearity in a saxophone. The reed is functioning as a switching diode, that rectifies the waveform, and so adds even harmonics to the sound of the instrument. The reed is operating between the mouth and a resonator, which colours the harmonic content.

It seems to me, that growling is actually a linear beat between the vocal, and the saxophone fundamental. That is why the growl frequency can be below both the voice and the fundamental. If the growl then interacts with the reed, it adds more harmonics.

I thought that as well but have proven that wrong by recording a spectrogram. If the growl were just a beat frequency, it wouldn't show up on the spectrogram but it does..

 

[Baluncore]

If the growl were just a beat frequency, it wouldn't show up on the spectrogram but it does.

If the growl comes from the saxophone, it involves the non-linearity of the reed, so energy can then appear at the beat frequency and the harmonics of that.

There is a parallel, from before the time of GPS. To transmit a master 1 kHz reference over a telephone line, the Frequency Division Multiplexing would introduce frequency errors. To overcome that drift problem, the 2'nd and 3'rd harmonics were generated and added, so they beat together, and sent in place of the 1 kHz. At the destination, the beat signal was passed through a diode, so a rock-solid 1 kHz difference frequency, could be extracted with a 1 kHz filter. Any FDM affected the harmonics equally, so the difference remained 1 kHz.

 

[Daniel Petka]

If the growl comes from the saxophone, it involves the non-linearity of the reed, so energy can then appear at the beat frequency and the harmonics of that.

There is a parallel, from before the time of GPS. To transmit a master 1 kHz reference over a telephone line, the Frequency Division Multiplexing would introduce frequency errors. To overcome that drift problem, the 2'nd and 3'rd harmonics were generated and added, so they beat together, and sent in place of the 1 kHz. At the destination, the beat signal was passed through a diode, so a rock-solid 1 kHz difference frequency, could be extracted with a 1 kHz filter. Any FDM affected the harmonics equally, so the difference remained 1 kHz.

Sure, it's all in the reed, the rest is basically just a tube. But even then I still don't understand why singing modulates the reed vibration..

 

[Baluncore]

But even then I still don't understand why singing modulates the reed vibration..

Where are you singing ?

 

[Vanadium 50]

I don't think you have to look far to find a non-linearity in a saxophone.

Except maybe soprano saxophones.

Pretty much every instrument has some non-linearity built in. The air column is coupled to the reed, as well as the body. Plenty of places for this to happen. Also, you have an impedance mismatch between the bell and the ouside world: most of the energy gets reflected back.

 

[Baluncore]

Also, you have an impedance mismatch between the bell and the ouside world: most of the energy gets reflected back.

The bell is designed to reduce the mismatch, to radiate sufficient sound, but it also provides a path for other sounds to enter the tube, that can then reach the reed.

 

[Daniel Petka]

I'm singing while playing, but it's not obvious to me at all why that should do anything. Take a flute for example - it doesn't work with a flute..

 

[Daniel Petka]

Except maybe soprano saxophones.
View attachment 338224
Pretty much every instrument has some non-linearity built in. The air column is coupled to the reed, as well as the body. Plenty of places for this to happen. Also, you have an impedance mismatch between the bell and the ouside world: most of the energy gets reflected back.

Why soprano saxophones? I did this on a clarinet btw

As far as I understand it the backreflections are what causes resonance itself, i.e. a standing wave within the cavity of the instrument. But I'm not sure if this is relevant. If thats the case, I should be able to get this to work on the mouthpiece alone

 

[Vanadium 50]

Why soprano saxophones?

It's a joke. A soprano is straight (well, there are curved ones, but they are not so popular) - as in "linear". Alto and tenors are curved, and baritones and bass saxes have a loop.

The bell is designed to reduce the mismatch, to radiate sufficient sound,

I would instead say the right amount of mismatch. You want the right amount of energy transmission from the internal standing wave to the external traveling wave. It;s surely higher for woodwinds, but for brasswinds, it's about 1% transmission and 99% reflection.

 

[Daniel Petka]

It's a joke. A soprano is straight (well, there are curved ones, but they are not so popular) - as in "linear". Alto and tenors are curved, and baritones and bass saxes have a loop.I would instead say the right amount of mismatch. You want the right amount of energy transmission from the internal standing wave to the external traveling wave. It;s surely higher for woodwinds, but for brasswinds, it's about 1% transmission and 99% reflection.

Haha r/woosh for me

Anyways, I found out that the whole resonator tube doesn't matter. I did it with just a mouthpiece and yeah it works.

The bottom spectrum comes from the mouthpiece (around 800Hz but not important), the one in the middle is my voice (falsetto obviously) and finally both combined. The subharmonic is still there. At this point it can't be called a growl, sounds more like a dying guinea-pig.
Maybe the speed of the reed vibration depends on the pressure. So an oscillating pressure can somehow excite that subharmonic

 

[Baluncore]

Maybe the speed of the reed vibration depends on the pressure. So an oscillating pressure can somehow excite that subharmonic

As far as I can see, the reed is the only contender for non-linearity in the saxophone. There is however a possibility of the human vocal cords, switching in a similar way. That may explain why beginners, learning to growl, suffer some pain.

Amplitude distortion generates higher frequency harmonics, f⋅n. Symmetry about zero makes odd harmonics, asymmetry about zero makes even harmonics.

Maybe the tendency to support the 'even saxophone' harmonics, or the 'odd clarinet' harmonics, are determined by a different symmetry of mounting and switching of the reed.

Sub-harmonic generation, of lower frequencies, f/n, requires energy or information storage. In other words, there must be an accumulation of something somewhere, that is then switched by the reed when it exceeds a critical threshold. That accumulation could be pressure or airflow in the mouthpiece, or the amplitude of a standing wave in the instrument, that presents a changing differential pressure across the reed after every two or three cycles.

 

[sophiecentaur]

mixing can only happen when there is some non-linear process involved

The motion of a reed is extremely non-linear (and in three dimensions, to boot) . The basic note from a reed instrument is full of harmonics and overtones. You can't treat the sounds from musical instruments as sine waves (except in a flute which has one of the purest output). Singing into the mouthpiece whilst playing a note will add pressure variations in the air and that will produce intermodulation products. The term "intermodulation" includes all products of nonlinearities when more than one fundamental signal is involved. Also you cannot ignore the difference between harmonics and overtones in any but the most idealised form of musical instrument.

My only experience of playing a wind instrument is when I bought a trumpet for £2 in 1965 but I well remember that it was easy to produce intermodulation products by singing / shouting / growling. That's hardly surprising for the same reasons as in a reed instrument. I just tried to get the same effect with a penny whistle that happens to be in my desk drawer. I don't have enough lung capacity these days to do a demo that would convince a skeptic about this so don't ask for a recording - sorry..

 

[Baluncore]

My friends and neighbours are still hoping, that one day I will finally master the kazoo.
https://en.wikipedia.org/wiki/Kazoo

 

[Daniel Petka]

The motion of a reed is extremely non-linear (and in three dimensions, to boot) . The basic note from a reed instrument is full of harmonics and overtones. You can't treat the sounds from musical instruments as sine waves (except in a flute which has one of the purest output). Singing into the mouthpiece whilst playing a note will add pressure variations in the air and that will produce intermodulation products. The term "intermodulation" includes all products of nonlinearities when more than one fundamental signal is involved. Also you cannot ignore the difference between harmonics and overtones in any but the most idealised form of musical instrument.

My only experience of playing a wind instrument is when I bought a trumpet for £2 in 1965 but I well remember that it was easy to produce intermodulation products by singing / shouting / growling. That's hardly surprising for the same reasons as in a reed instrument. I just tried to get the same effect with a penny whistle that happens to be in my desk drawer. I don't have enough lung capacity these days to do a demo that would convince a skeptic about this so don't ask for a recording - sorry..

I totally can treat them as a superposition of sine waves, though. (Fourier) what do you mean with "nonlinear motion". The system (reed oscillator) can be nonlinear, but not its motion, right? The easiest model for a nonlinear system I could find is the Duffing equation and the whole nonlinearity is in the cubic term. But there is no pressure dependence in that model...

Thanks for the idea, I might 3D print a penny whistle and try to growl on it.

 

[Daniel Petka]

As far as I can see, the reed is the only contender for non-linearity in the saxophone. There is however a possibility of the human vocal cords, switching in a similar way. That may explain why beginners, learning to growl, suffer some pain.

Amplitude distortion generates higher frequency harmonics, f⋅n. Symmetry about zero makes odd harmonics, asymmetry about zero makes even harmonics.

Maybe the tendency to support the 'even saxophone' harmonics, or the 'odd clarinet' harmonics, are determined by a different symmetry of mounting and switching of the reed.

Sub-harmonic generation, of lower frequencies, f/n, requires energy or information storage. In other words, there must be an accumulation of something somewhere, that is then switched by the reed when it exceeds a critical threshold. That accumulation could be pressure or airflow in the mouthpiece, or the amplitude of a standing wave in the instrument, that presents a changing differential pressure across the reed after every two or three cycles.

Nope the even/odd harmonics have everything to do with the resonator (clarinet = tube, sax = cone) It's not really relevant for this discussion, but here is a video on it if you're interested:
But does this accumulation explain that I can

 

[Baluncore]

But does this accumulation explain that I can

That you "can" what, generate a sub-harmonic like a growl?
Please be more precise to eliminate misinterpretation.

I am still trying to model the extended system needed to growl. I am concerned that it may be your vocal cords, providing the non-linear mechanism. Or that your body somehow provides the longer resonator needed to support the growl.

I have insufficient experience or information to form and analyse the unknown system.
I just know that there must be a non-linear switch somewhere, with a storage in time, to generate energy at sub-harmonic frequencies.

It might help if you could identify what hurts, when you are learning to growl through a saxophone?

 

[Daniel Petka]

Nothing hurts at all when I growl on the clarinet. Sure, I do feel my chest vibrating as well, but like I mentioned in my previous post, resonance is completely irrelevant as you can do this on just a mouthpiece. It makes then no difference if the mouthpiece sits on a clarinet or a saxophone.

I came across the air flow vs pressure difference (between mouthpiece and mouth cavity) function of the clarinet and it's nonlinear in itself, something like:

$Flow(\Delta p) \propto (1-\Delta p) \sqrt{\Delta p}$

The peak of the plot is the threshold for playing. So in the end, you play at some operating point and yeah large oscillations around that point will be nonlinear. So in the end, I think Berkeman was right. The thing is I can never play quietly enough to produce a single sinusoid

 

[sophiecentaur]

I totally can treat them as a superposition of sine waves, though

That's not actually true. The natural resonances in musical instruments are seldom exact harmonics of a fundamental so you can't assume that harmonic analysis is appropriate. Wind instruments all suffer from the 'end effect' which affects the equivalent length of the tube for each frequency. If you take a length of copper plumbing pipe and produce a note on it (easiest with a suitable mouthpiece) the fundamental and overtones sound complete 'wrong' when you try to blow arpeggios (c g c e g a c etc) Listen to notes from a long hunting horn; a ghastly and unmusical sound - even if you're not a fox!! A skilled player will pull the notes to make them sound right but the timbre of the note is governed by this end effect, whatever you do. This accounts for the colouration of sounds from different instruments. Take a cornet (cylindrical bore) and a trumpet (conical bore) and compare the sounds. The trumpet is much more punchy than the smoother cornet note. Etc etc.

I feel that I must be a lone voice in the wilderness when I point this out but the Fourier Transform is by no means always appropriate when physical dimensions are involved (which is more often that you might think.) Modes, not harmonics apply in so many cases.

 

[Baluncore]

Sure, I do feel my chest vibrating as well, but like I mentioned in my previous post, resonance is completely irrelevant as you can do this on just a mouthpiece.

Which sort of confirms my suspicion that, the non-linearity is your vocal cords, and your chest is the resonator. Maybe we have been looking in the saxophone for the growl generator, when all the time it has been in the player.

 

[Daniel Petka]

Which sort of confirms my suspicion that, the non-linearity is your vocal cords, and your chest is the resonator. Maybe we have been looking in the saxophone for the growl generator, when all the time it has been in the player.

No, it doesn't confirm that. Like mentioned above, you can do this on a single mouthpiece. The nonlinearity is in the reed (see plot above)

 

[Baluncore]

Like mentioned above, you can do this on a single mouthpiece. The nonlinearity is in the reed (see plot above)

If you blow a single mouthpiece, is your vocal tract not on one side of that reed, and does your body not form a resonant cavity?

 

[Daniel Petka]

That's not actually true. The natural resonances in musical instruments are seldom exact harmonics of a fundamental so you can't assume that harmonic analysis is appropriate. Wind instruments all suffer from the 'end effect' which affects the equivalent length of the tube for each frequency. If you take a length of copper plumbing pipe and produce a note on it (easiest with a suitable mouthpiece) the fundamental and overtones sound complete 'wrong' when you try to blow arpeggios (c g c e g a c etc) Listen to notes from a long hunting horn; a ghastly and unmusical sound - even if you're not a fox!! A skilled player will pull the notes to make them sound right but the timbre of the note is governed by this end effect, whatever you do. This accounts for the colouration of sounds from different instruments. Take a cornet (cylindrical bore) and a trumpet (conical bore) and compare the sounds. The trumpet is much more punchy than the smoother cornet note. Etc etc.

I feel that I must be a lone voice in the wilderness when I point this out but the Fourier Transform is by no means always appropriate when physical dimensions are involved (which is more often that you might think.) Modes, not harmonics apply in so many cases.

Sure, harmonics and modes (eigenfunctions) are only identical if the instrument is a 1 dimensional string and a clarinet gets close to that. But in the end the sound you hear is in fact a sum of sine waves.

 

[sophiecentaur]

No, it doesn't confirm that. Like mentioned above, you can do this on a single mouthpiece. The nonlinearity is in the reed (see plot above)

Hmm. It's not at all clear cut. Agreed; you have a number of resonators and time constants involved. If you 'feel' a note in your chest then you may be hearing a resonance but only if the resonator you're suggesting is narrow band. The structure of the airways would probably have a very low Q factor with lots of damping. The vocal cords would probably be the resonators but, as with the reed, the basic audio spectrum is very dirty (no sine wave) so the interaction between reed and cords would not be simple.

 

[Daniel Petka]

If you blow a single mouthpiece, is your vocal tract not on one side of that reed, and does your body not form a resonant cavity?

Yes of course it's a resonator but what I'm saying is that resonance doesn't explain the modulation. A proof of this is that I can create a continuum of difference tones by changing my pitch. Sure, some will resonate and be louder as a consequence. The cavity size can change when you alter a pitch but doesn't have to.

After all, there are plenty of linear resonators out there.

 

[Baluncore]

Yes if course it's a resonator but what I'm saying is that resonance doesn't explain the modulation.

The reed provides the non-linear switching required for sub-harmonic generation.
The reed is situated between two cavities, and so is subjected to a differential pressure.
The energy storage can then be in the body, with the vocal cords that produce the tone.
The combined beat passes through the reed, moving energy to the sub-harmonic.

 

[Daniel Petka]

Yes, this sounds right. All the nonlinearity is in the reed.

 

[sophiecentaur]

But in the end the sound you hear is in fact a sum of sine waves.

Of course but they're not harmonically related, necessarily - is my point.

does your body not form a resonant cavity?

At what resonant frequency? If the Q factor is low then it's not relevant and will not affect the amplitude.

 

[Daniel Petka]

I'm singing while playing, but it's not obvious to me at all why that should do anything. Take a flute for example - it doesn't work with a flute..

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

 

[tech99]

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

The reed is a relaxation oscillator, so the frequency is pressure sensitive. The second tone apples pressure waves to the reed. So my suggestion is that the second tone frequency modulates the original tone, giving rise to additional side frequencies.

 

[Daniel Petka]

The reed is a relaxation oscillator, so the frequency is pressure sensitive. The second tone apples pressure waves to the reed. So my suggestion is that the second tone frequency modulates the original tone, giving rise to additional side frequencies.

Yes, it's modulation, as I have mentioned in my post and it's caused by the reed's nonlinearity. More pressure inside the mouth means more are comes out, which it turn decreases the pressure and closes the reed. The cycle repeats because the pressure wave gets backreflected at the end of the pipe. The pressure vs air plot shows a curved slope, and when you sing a note you basically move the operating point around which you oscillate. I have yet to figure out how the flutes linearity gets created. Another rabbithole, yay

 

[sophiecentaur]

The reed is a relaxation oscillator, so the frequency is pressure sensitive.

Yes, exactly. The "resonances" that have been mentioned provide broad band matching rather than frequency selection. The bell on the end of many wind instruments is for matching and, at the same time, it pulls the frequencies of the overtones away from where they'd be with a straight cut end.

The reed provides the non-linear switching required for sub-harmonic generation.
The reed is situated between two cavities, and so is subjected to a differential pressure.
The energy storage can then be in the body, with the vocal cords that produce the tone.
The combined beat passes through the reed, moving energy to the sub-harmonic.

I don't see how low frequency product generation is any more relevant than high frequency products. If we're discussing steady state situations, any relative delay in formation of products doesn't count. During the initial formation of a note, the attack waveform will always be different because of the initial relative delays. A plucked string is a good example of where the attack portion is often the most important bit.

Many discussions about musical instruments spend too much time dealing with the steady state situation and 'Harmonics'. Far too simple for an understanding of what goes on. Early electric / electronic organs had very distinctive sounds; nothingl lke the suggested instrument names on the key switches. But the Hammond sound is known and loved for its artificiality.